I have got my rectangular grid in Adobe Illustrator CS3 on Mac OS X Lion 10.7.3:
How do I make it so that it puts a shape/dot/object on the intersections, where the lines meet each other?
You need enable the Snap Option, and Smart Guides.
In your case, Snap to Point is the case (View > Snap to Point) . And the Smart Guides (View > Smart Guides)
[I am using Illustrator CS5, but Im sure these have been there for a while, not sure since when]